耦合振子同步的最优网络设计
Optimal network design for synchronization of coupled oscillators
摘要：
本文研究的问题，设计网络的不同的耦合振荡器，以达到所需的水平的相位凝聚力，定义为最大渐近相位差的边缘的网络。特别是，我们考虑以下两个设计问题：（i）频率设计问题，其中我们调整振荡器的固有频率给定网络的拓扑结构，（ii）（鲁棒）边权重设计问题，其中我们设计边权重假设固有频率给定（或属于给定的凸不确定性集）。对于这两个问题，我们优化的目标函数的设计变量，同时考虑所需水平的相位内聚性作为我们的设计约束。该约束在中频设计问题中定义了一个凸集。相比之下，在边权重设计问题中，相位内聚性约束产生非凸集，除非底层网络是具有相同边权重的树或任意图。然后，我们提出了一个凸半定松弛近似解决（非凸）边权重设计问题的一般（可能是循环）网络与不相同的边权重。我们说明了我们的研究结果的适用性，通过分析几个网络设计问题的实际利益，如电力再调度电网，稀疏网络设计，（鲁棒）网络设计分布式无线模拟时钟，并检测边缘导致Braess'悖论在电网中。
英文摘要：
This paper studies the problem of designing networks of nonidentical coupled oscillators in order to achieve a desired level of phase cohesiveness, defined as the maximum asymptotic phase difference across the edges of the network. In particular, we consider the following two design problems: (i) the nodal-frequency design problem, in which we tune the natural frequencies of the oscillators given the topology of the network, and (ii) the (robust) edge-weight design problem, in which we design the edge weights assuming that the natural frequencies are given (or belong to a given convex uncertainty set). For both problems, we optimize an objective function of the design variables while considering a desired level of phase cohesiveness as our design constraint. This constraint defines a convex set in the nodal-frequency design problem. In contrast, in the edge-weight design problem, the phase cohesiveness constraint yields a non-convex set, unless the underlying network is either a tree or an arbitrary graph with identical edge weights. We then propose a convex semidefinite relaxation to approximately solve the (non-convex) edge-weight design problem for general (possibly cyclic) networks with nonidentical edge weights. We illustrate the applicability of our results by analyzing several network design problems of practical interest, such as power re-dispatch in power grids, sparse network design, (robust) network design for distributed wireless analog clocks, and the detection of edges leading to the Braess' paradox in power grids.
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